Fluid Physics
8.292J/12.330J
Problem Set 2
1. A shallow layer of Euler fluid of depth 
in a uniform
gravitational field, given by acceleration 
, flows from left to right with velocity 
and encounters a smooth bump of height 
. The bottom returns to its initial height rightward of the
bump. Consider the flow to be two-dimensional (i.e. without variation in the
horizontal direction normal to the flow). With one exception noted below, the
flow can be considered steady. We
define a nondimensional number 
(called the Froude
number):
The
flow can assume one of two configurations, as illustrated below:
(a)
(b)
In
regime (b), a hydraulic jump (a stationary bore) occurs downstream of the bump;
the flow near it is turbulent and dissipative.
a.)
For each of the two regimes, find
expressions for 
and 
.
b.) For regime (b), also find expressions for 
and 
.
c.) Find approximate expressions for 
and 
in the limit that 
is very small.
d.) Find approximate expressions for 
and 
in the limit that 
is very large.
e.) Extra credit: For regime (b), find an
expression for the net horizontal force on the bump, being sure to indicate
which direction it acts.
2a. A barge, floating on a pond, is equipped with a pump which takes
water from the pond and pumps it out through a hose fitted with a nozzle (a).
The water may be idealized as an Euler fluid for this problem. The density of
the water is 
, the cross-sectional area of the end of the nozzle is 
, and the velocity of the water as it exits the nozzle is 
. Find the force on the barge.
(a)
2b. Consider the same problem
as in 2a, but this time the pump is located on a dock (fixed to the ground; see
(b)). For the purposes of this problem you may neglect gravity and assume
that there are no tensile forces acting on the hose. What is the force on the
barge? How might your answer change if you allow for friction of the water
flowing through the hose?
(b)
3.
(Extra
credit) Consider the problem described
in problem 2b, but allow for gravity, so that the hose sags. Assume that the
hose has infinite tensile strength (it cannot be compressed or extended along
its length) but has no resistance to bending. Neglect friction of the flowing
water. What is the force on the barge?
(Note: to get a quantitative answer, you might want to consider special
configurations of the hose.)